Lattice and Sublattice

Question

I understand the definition of Lattice and Sublattice, but I cannot show the following exercise, since it seems trivial to me.

Show that, for any subset $S$ of a lattice $L$, the set $s^{l}$ of all the lower bounds of $S$ is a sublattice of $L$.

Thanks in advance.

Answer 1

It's indeed easy: if $a$ and $b$ are lower bounds for all $s\in S$, that means $a\le s, \, b\le s$.
Then clearly also $a\land b\le s$ as e.g. $a\land b\le a$, and $a\lor b\le s$ by the definition of least upper bound ($\lor$).